Question: All of the 5th grade teachers and students from Oak went on a field trip to an archaeology museum. Tickets were $$6.50$ each for teachers and $$2.50$ each for students, and the group paid $$44.50$ in total. The next month, the same group visited a science museum where the tickets cost $$13.00$ each for teachers and $$8.00$ each for students, and the group paid $$119.00$ in total. Find the number of teachers and students on the field trips.
Let $x$ equal the number of teachers and $y$ equal the number of students. The system of equations is: ${6.5x+2.5y = 44.5}$ ${13x+8y = 119}$ Solve for $x$ and $y$ using elimination. Multiply the top equation by $-2$ ${-13x-5y = -89}$ ${13x+8y = 119}$ Add the top and bottom equations together. $ 3y = 30 $ $ y = \dfrac{30}{3}$ ${y = 10}$ Now that you know ${y = 10}$ , plug it back into $ {6.5x+2.5y = 44.5}$ to find $x$ ${6.5x + 2.5}{(10)}{= 44.5}$ $6.5x+25 = 44.5$ $6.5x = 19.5$ $x = \dfrac{19.5}{6.5}$ ${x = 3}$ You can also plug ${y = 10}$ into $ {13x+8y = 119}$ and get the same answer for $x$ ${13x + 8}{(10)}{= 119}$ ${x = 3}$ There were $3$ teachers and $10$ students on the field trips.